Justin Tatch Moore
In this note we will discuss a new reflection principle which follows from the Proper Forcing Axiom. The immediate purpose will be to prove that the bounded form of the Proper Forcing Axiom implies both that 2^omega = omega_2 and that L(P(omega_1)) satisfies the Axiom of Choice. It will also be demonstrated that this reflection principle implies that combinatorial principle Square(kappa) fails for all regular kappa > omega_1.
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We study projective stationary sets. The Projective Stationary Reflection principle is the statement that every projective stationary set contains an increasing continuous $\in$--chain of length $ω_1$. We show that if Martin's Maximum holds, then the Projective Stationary Reflection Principle holds. Also it is equivalent to the Strong Reflection Principle. We show that the saturation of the nonstationary ideal on $ω_1$ is equivalent to a certain kind of reflection.
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set r…
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
We show that induction over $Δ(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.
Forcing axioms, approachability, and stationary set reflection
本文证明平稳集反射与内部可逼近性的多种定理，指出Fuchino-Usuba蕴含不可逆，加强并简化Krueger结果，并借鉴Woodin思想统一力迫公理保持论证。
It is consistent that for every n >= 2, every stationary subset of omega_n consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects fully in the set of ordinals of cofinality omega_{n-1}. We also show that this result is best possible.
We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.
We reconsider here the following related pcf questions and make some advances: (Q1) concerning the ideal {check I}_kappa [lambda] how much reflection do we have for the bad set S^{bd}_{lambda, kappa} subseteq {delta < lambda : cf(delta)= kappa} assuming it is well defined? (Q2) for an ideal J on kappa how large are S^{bd}_J[f],S^{ch}_J[f] for f=< f_alpha : alpha<lambda > which is <_J-increasing and cofinal in (prod limits_{i< kappa} lambda_i,<_J) ? (Q3) are there somewhat free black boxes?
Generalized ordinal analysis and reflection principles in set theory
针对自然公理系统良序层级的质疑，本文基于集合论反射原理提出证明论强度概念，证明其与Pohlers广义序数分析的理论序一致，生成良序层级，并推广至其他内模型。
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.
A stationary subset $S$ of a regular uncountable cardinal $κ$ {\it reflects fully} at regular cardinals if for every stationary set $T \subseteq κ$ of higher order consisting of regular cardinals there exists an $α\in T$ such that $S \cap α$ is a stationary subset of $α$. {\it Full Reflection} states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $κ$ having Mitchell order $κ^{++}$ it is consistent that Full Reflection holds at every $λ\leq κ$ and $κ$ is measurable.
Reflection in second-order set theory with abundant urelements bi-interprets a supercompact cardinal
本文探讨了带原子的二阶集合论中二阶反射的强度，证明了带丰富原子公理的二阶反射与超紧致基数存在性可互译且等一致。
Separating diagonal stationary reflection principles
本文引入三族对角反射原理，分析其相互关系及与已知原理的联系。
We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.
We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $φ(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $φ(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy-Montague reflection theorem. They are…
The set theory KP$Π_{N+1}$ for $Π_{N+1}$-reflecting universes is shown to be $Π_{N+1}$-conservative over iterations of $Π_{N}$-recursively Mahlo operations for each $N\geq 2$.
section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)
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